Again, no. The center of thrust does move forward, but it also moves up compared to the 737NG, so the thrust vector actually ends up closer to the center of mass. And center of thrust changes affect how the aircraft handles based on throttle response, not on pitch inputs. The engine nacelles matter, but due to the center of lift changing position with angle of attack, so that at high angles of attack it’s further forward than usual. This isn’t to say that the CoL moves ahead of the CoM. The aircraft remains stable at all times pre-stall.
Yes, there’s a controllability issue, but it’s at near stall conditions. It’s not going to cause an issue in normal flight conditions, as pilots will universally avoid getting their aircraft in near stall conditions while carrying passengers. FBW wouldn’t be necessary to damp out the oscillation. Rather, because all stick forces are entirely synthetic in FBW, they’d just generate the appropriate amount of stick force to keep the force curve in compliance.
It still is.